On the Oscillation Criteria and Computation of Third Order Oscillatory Differential Equations

Authors

  • J. Sunday Department of Mathematics, Adamawa State University, Mubi

DOI:

https://doi.org/10.26713/cma.v9i4.968

Keywords:

Analysis, Computation, Bounded solutions, Infinite zeros, Oscillation, Third-order

Abstract

One of the sets of differential equations that find applications in real life is the oscillatory problems. In this paper, the oscillation criteria and computation of third order oscillatory differential equations are studied. The conditions for a third order differential equation to have oscillatory solutions on the interval \(I=[t_0,\infty)\)  shall be analyzed. Further, a highly efficient and reliable one-step computational method (with three partitions) is formulated for the approximation of third order differential equations. The paper also analyzed the basic properties of the method so formulated. The results obtained on the application of the method on some sampled modeled third order oscillatory problems show that the method is computationally reliable and the method performed better than the ones with which we compared our results.

Downloads

Download data is not yet available.

References

A.O. Adesanya, M.A. Alkali and J. Sunday, Order five hybrid block method for the solution of second order ordinary differential equations, International J. Math. Sci. & Engg. Appls. 8(3) (2014), 285 – 295.

A.O. Adesanya, D.M. Udoh and A.M. Ajileye, A new hybrid block method for the solution of general third order initial value problems of ODEs, International Journal of Pure and Applied Mathematics 86(2) (2013), 365 – 375, DOI: 10.12732/ijpam.v86i2.11.

A.O. Adesanya, O.M. Udoh and A.M. Alkali, A new block-predictor corrector algorithm for the solution of (y'''= (x, y, y', y'')), American Journal of Computational Mathematics 2 (2012), 341 – 344, DOI: 10.4236/ajcm.2012.24047.

D.O. Awoyemi, S.J. Kayode and L.O. Adoghe, A five-step P-stable method for the numerical integration of third order ordinary differential equations, American Journal of Computational Mathematics 4 (2014), 119 – 126, DOI: 10.4236/ajcm.2014.43011.

E.J. Borowski and J.M. Borwein, Dictionary of Mathematics, Harper Collins Publishers, Glasgow (2005).

G.G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Mathematica Scandinavica 4 (1956), 33 – 50.

B.R. Duffy and S.K. Wilson, A third order differential equation arising in thin-film flows and relevant to Tanner's law, Appl. Math. Lett. 10(3) (1997), 63 – 68.

S.O. Fatunla, Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Academic Press Inc, New York (1988).

R. Genesio and A. Tesi, Harmonic balance methods for the alanysis of chaotic dynamics in nonlinear systems, Antomatica 28(3) (1992), 531 – 548.

M. Hanan, Oscillation criteria for third order linear differential equations, Pacific Journal of Mathematics 11(3) (1961), 919 – 944.

B. Kanat, Numerical Solution of highly oscillatory differential equations by magnus series method, Unpublished master's thesis, Izmir Institute of Technology, Izmir (2006).

J.D. Lambert, Numerical methods for ordinary differential systems: The Initial Value Problem, John Willey and Sons, New York (1991).

K.Y. Lee, I. Fudziah and S. Norazak, An accurate block hybrid collocation method for third order ordinary differential equations, Journal of Applied Mathematics (2014), 1 – 9, DOI: 10.1155/2014/549597.

A.Z. Majid, N.A. Azni, M. Suleiman and B.I. Zarina, Solving directly general third order ordinary differential equations using two-point four step block method, Sains Malaysiana 41(5) (2012), 623 – 632.

U. Mohammed and R.B. Adeniyi, A three step implicit hybrid linear multistep method for the solution of third order ordinary differential equations, Gen. Math. Notes 25(1) (2014), 62 – 74.

J. Sunday, The Duffing oscillator: applications and computational simulations, Asian Research Journal of Mathematics 2(3) (2017), 1 – 13, DOI: 10.9734/ARJOM/2017/31199.

H.A. Stetter, Development of ideas, techniques and implementation, Proceeding of Symposium in Applied Mathematics 48 (1994), 205 – 224.

N.S. Yakusak, S.T. Akinyemi and I.G. Usman, Three off-steps hybrid method for the numerical solution of third order initial value problems, IOSR Journal of Mathematics 12(3) (2016), 59 – 65, iosrjournals.org/iosr-jm/Vo.12-issue3/Version-2/I1203025965.pdf.

Y.L. Yan, Numerical Methods for Differential Equations, City University of Hong Kong, Kowloon (2011).

Downloads

Published

25-12-2018
CITATION

How to Cite

Sunday, J. (2018). On the Oscillation Criteria and Computation of Third Order Oscillatory Differential Equations. Communications in Mathematics and Applications, 9(4), 615–626. https://doi.org/10.26713/cma.v9i4.968

Issue

Vol. 9 No. 4 (2018)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.